Question: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{k^2 - 36}{k + 6}$
Answer: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = k$ $ b = \sqrt{36} = 6$ So we can rewrite the expression as: $z = \dfrac{({k} + {6})({k} {-6})} {k + 6} $ We can divide the numerator and denominator by $(k + 6)$ on condition that $k \neq -6$ Therefore $z = k - 6; k \neq -6$